#9224189

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4860 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,5,8)(3,6,7,4)(17,18)(19,24)(20,22)(21,25)(23,27)(26,32)(28,30)(29,31);(1,2,5,8)(3,4,7,6)(9,10)(11,12)(13,15)(14,16)(17,19)(18,23)(20,24)(21,26)(22,27)(25,31)(28,32)(29,30);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16);(3,7)(4,6)(9,12,13,14)(10,11,15,16)(17,20,22,18)(19,24,27,23)(21,25,30,28)(26,31,29,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,16)(14,15)(17,19)(18,24)(20,23)(21,29)(22,27)(25,31)(26,30)(28,32);(1,4,5,6)(2,7,8,3)(9,10)(11,14)(12,16)(13,15)(17,21,22,30)(18,25,20,28)(19,26,27,29)(23,31,24,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9131385);